A Simple Reform for Treating the Loss of Accuracy of Humlíček’s W4 Algorithm Near the Real Axis

Mofreh Ramadan Zaghloul

doi:doi:10.11648/j.acm.20221102.13

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A Simple Reform for Treating the Loss of Accuracy of Humlíček’s W4 Algorithm Near the Real Axis

Mofreh Ramadan Zaghloul

Published in Applied and Computational Mathematics (Volume 11, Issue 2)

Received: 20 March 2022 Accepted: 6 April 2022 Published: 25 April 2022

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Abstract

Rapid evaluation of the Faddeyeva function, also known as the complex probability function, is essential to many spectroscopic and stellar applications. Humlíček’s W4 Algorithm is widely used in the literature for rapid and marginally accurate evaluation of the function (~10^-4). However, as reported in the literature, the algorithm lose its claimed accuracy near the x-axis. In this paper, we present a simple reform for treating the reported problem of loss-of-accuracy near the real axis of the algorithm. The reform is reached through region-borders rearrangement which is reflected as a very minor coding change to the original w4 algorithm that can be straightforwardly implemented. The reformed routine maintains the claimed accuracy of the algorithm over a wide and fine grid that covers all the domain of the real part, x, of the complex input variable, z=x+iy, and values for the imaginary part in the range y=Î [10^-30, 10³⁰].

Published in	Applied and Computational Mathematics (Volume 11, Issue 2)
DOI	10.11648/j.acm.20221102.13
Page(s)	56-59
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Faddeyeva Function, Complex Probability Function, Voigt Function, W4 Algorithm

References

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Mofreh Ramadan Zaghloul. (2022). A Simple Reform for Treating the Loss of Accuracy of Humlíček’s W4 Algorithm Near the Real Axis. Applied and Computational Mathematics, 11(2), 56-59. https://doi.org/10.11648/j.acm.20221102.13

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Mofreh Ramadan Zaghloul. A Simple Reform for Treating the Loss of Accuracy of Humlíček’s W4 Algorithm Near the Real Axis. Appl. Comput. Math. 2022, 11(2), 56-59. doi: 10.11648/j.acm.20221102.13

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AMA Style

Mofreh Ramadan Zaghloul. A Simple Reform for Treating the Loss of Accuracy of Humlíček’s W4 Algorithm Near the Real Axis. Appl Comput Math. 2022;11(2):56-59. doi: 10.11648/j.acm.20221102.13

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@article{10.11648/j.acm.20221102.13,
  author = {Mofreh Ramadan Zaghloul},
  title = {A Simple Reform for Treating the Loss of Accuracy of Humlíček’s W4 Algorithm Near the Real Axis},
  journal = {Applied and Computational Mathematics},
  volume = {11},
  number = {2},
  pages = {56-59},
  doi = {10.11648/j.acm.20221102.13},
  url = {https://doi.org/10.11648/j.acm.20221102.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20221102.13},
  abstract = {Rapid evaluation of the Faddeyeva function, also known as the complex probability function, is essential to many spectroscopic and stellar applications. Humlíček’s W4 Algorithm is widely used in the literature for rapid and marginally accurate evaluation of the function (~10-4). However, as reported in the literature, the algorithm lose its claimed accuracy near the x-axis. In this paper, we present a simple reform for treating the reported problem of loss-of-accuracy near the real axis of the algorithm. The reform is reached through region-borders rearrangement which is reflected as a very minor coding change to the original w4 algorithm that can be straightforwardly implemented. The reformed routine maintains the claimed accuracy of the algorithm over a wide and fine grid that covers all the domain of the real part, x, of the complex input variable, z=x+iy, and values for the imaginary part in the range y=Î [10-30, 1030].},
 year = {2022}
}

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TY  - JOUR
T1  - A Simple Reform for Treating the Loss of Accuracy of Humlíček’s W4 Algorithm Near the Real Axis
AU  - Mofreh Ramadan Zaghloul
Y1  - 2022/04/25
PY  - 2022
N1  - https://doi.org/10.11648/j.acm.20221102.13
DO  - 10.11648/j.acm.20221102.13
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 56
EP  - 59
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20221102.13
AB  - Rapid evaluation of the Faddeyeva function, also known as the complex probability function, is essential to many spectroscopic and stellar applications. Humlíček’s W4 Algorithm is widely used in the literature for rapid and marginally accurate evaluation of the function (~10-4). However, as reported in the literature, the algorithm lose its claimed accuracy near the x-axis. In this paper, we present a simple reform for treating the reported problem of loss-of-accuracy near the real axis of the algorithm. The reform is reached through region-borders rearrangement which is reflected as a very minor coding change to the original w4 algorithm that can be straightforwardly implemented. The reformed routine maintains the claimed accuracy of the algorithm over a wide and fine grid that covers all the domain of the real part, x, of the complex input variable, z=x+iy, and values for the imaginary part in the range y=Î [10-30, 1030].
VL  - 11
IS  - 2
ER  -

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Author Information

Mofreh Ramadan Zaghloul

Department of Physics, College of Sciences, United Arab Emirates University, AlAin, United Arab Emirates

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Mofreh Ramadan Zaghloul. (2022). A Simple Reform for Treating the Loss of Accuracy of Humlíček’s W4 Algorithm Near the Real Axis. Applied and Computational Mathematics, 11(2), 56-59. https://doi.org/10.11648/j.acm.20221102.13

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ACS Style

Mofreh Ramadan Zaghloul. A Simple Reform for Treating the Loss of Accuracy of Humlíček’s W4 Algorithm Near the Real Axis. Appl. Comput. Math. 2022, 11(2), 56-59. doi: 10.11648/j.acm.20221102.13

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AMA Style

Mofreh Ramadan Zaghloul. A Simple Reform for Treating the Loss of Accuracy of Humlíček’s W4 Algorithm Near the Real Axis. Appl Comput Math. 2022;11(2):56-59. doi: 10.11648/j.acm.20221102.13

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@article{10.11648/j.acm.20221102.13,
  author = {Mofreh Ramadan Zaghloul},
  title = {A Simple Reform for Treating the Loss of Accuracy of Humlíček’s W4 Algorithm Near the Real Axis},
  journal = {Applied and Computational Mathematics},
  volume = {11},
  number = {2},
  pages = {56-59},
  doi = {10.11648/j.acm.20221102.13},
  url = {https://doi.org/10.11648/j.acm.20221102.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20221102.13},
  abstract = {Rapid evaluation of the Faddeyeva function, also known as the complex probability function, is essential to many spectroscopic and stellar applications. Humlíček’s W4 Algorithm is widely used in the literature for rapid and marginally accurate evaluation of the function (~10-4). However, as reported in the literature, the algorithm lose its claimed accuracy near the x-axis. In this paper, we present a simple reform for treating the reported problem of loss-of-accuracy near the real axis of the algorithm. The reform is reached through region-borders rearrangement which is reflected as a very minor coding change to the original w4 algorithm that can be straightforwardly implemented. The reformed routine maintains the claimed accuracy of the algorithm over a wide and fine grid that covers all the domain of the real part, x, of the complex input variable, z=x+iy, and values for the imaginary part in the range y=Î [10-30, 1030].},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - A Simple Reform for Treating the Loss of Accuracy of Humlíček’s W4 Algorithm Near the Real Axis
AU  - Mofreh Ramadan Zaghloul
Y1  - 2022/04/25
PY  - 2022
N1  - https://doi.org/10.11648/j.acm.20221102.13
DO  - 10.11648/j.acm.20221102.13
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 56
EP  - 59
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20221102.13
AB  - Rapid evaluation of the Faddeyeva function, also known as the complex probability function, is essential to many spectroscopic and stellar applications. Humlíček’s W4 Algorithm is widely used in the literature for rapid and marginally accurate evaluation of the function (~10-4). However, as reported in the literature, the algorithm lose its claimed accuracy near the x-axis. In this paper, we present a simple reform for treating the reported problem of loss-of-accuracy near the real axis of the algorithm. The reform is reached through region-borders rearrangement which is reflected as a very minor coding change to the original w4 algorithm that can be straightforwardly implemented. The reformed routine maintains the claimed accuracy of the algorithm over a wide and fine grid that covers all the domain of the real part, x, of the complex input variable, z=x+iy, and values for the imaginary part in the range y=Î [10-30, 1030].
VL  - 11
IS  - 2
ER  -

Copy | Download